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© ESO 2019. The aim of this paper is to point out that the classic seismological problem using observations and theoretical expressions for the periods and damping times of transverse standing magnetohydrodynamic waves in coronal loops is better referred to as a reduced seismological problem. "Reduced" emphasises the fact that only a small number of characteristic quantities of the equilibrium profiles can be determined. Reduced also implies that there is no unique solution to the full seismological problem. Even the reduced seismological problem does not allow a unique solution. Bayesian inference results support our mathematical arguments and offer insight into the relationship between the algebraic and the probabilistic inversions. ispartof: ASTRONOMY & ASTROPHYSICS vol:622 status: published

We study the excitation and damping of transverse oscillations in a complex multi-stranded model of a coronal loop. By numerically solving the time-dependent magnetohydrodynamic (MHD) equations in two dimensions, we show how the global motion of the whole bundle of tubes, produced by an external disturbance, is converted into localised motions due to the process of resonant absorption. At any location in the structure two dominant frequencies are found, the frequency of the global mode (different from the kink frequency of the individual strands) and the local Alfvén frequency. The mechanism of mode conversion is not affected by the complicated geometry of the system and for certain configurations the energy conversion does not only take place at the external edge of the composite loop but also inside the structure.

We study the effects of magnetic field shear (B y ¬= 0) and longitudinal propagation of perturbations (k y ¬= 0) on the linear and adiabatic magnetohydrodynamic (MHD) normal modes of oscillation of a potential coronal arcade. In a cold plasma, the inclusion of these two effects produces the linear coupling of discrete fast modes, characterised by a discrete spectrum of frequencies and a global velocity structure, and Alfven continuum modes, characterised by a continuous spectrum of frequencies and with a velocity perturbation confined to given magnetic surfaces in such a way that modes with mixed properties arise (Arregui et al. 2004). The wave equations governing the velocity perturbations have been solved numerically and our results show that the couplings between fast and Alfven modes are governed by some parity rules for the symmetry of the eigenfunctions of fast and Alfven modes in the direction along the equilibrium magnetic field. The nature of the coupling between fast and Alfven modes can be resonant or non-resonant depending on the location of the fast mode frequency within the different Alfven continua. Also, an important result is that in this kind of configurations coupled modes could be difficult to observe since when both magnetic field shear and longitudinal propagation are present the spatial distribution of the velocity may not be confined to low heights in the solar corona.

The theoretical understanding of the linear standing or propagating magnetohydrodynamic waves in a variety of solar coronal structures is far from complete since analytical solutions to the linearised MHD equations can only be found for very simple magnetic configurations. In this paper, we use a numerical code to solve the linear fast and Alfven wave equations in a very simple, bounded magnetic configuration that incorporates two features that are not usually considered in similar works, namely the longitudinal magnetic field component and wave propagation in the longitudinal direction ($k_y\neq 0$). We use a numerical code (Arregui et al. 2001) that has been modified by including a staggered mesh that allows us to properly capture the spatial behaviour of solutions to the wave equations. Coupling between fast and Alfven modes has been studied in detail and it has been found that it does not take place when the longitudinal field component is zero and the frequency of the fast mode is outside the Alfven continuum with the same spatial structure along field lines. Under these circumstances, fast modes retain their global spatial behaviour and are also characterised by $\omega^2$ varying linearly with $k_y^2$, such as in a uniform medium (although here the Alfven speed changes exponentially in the direction normal to field lines). Regarding mode coupling, its main feature is the blend of fast and Alfven solutions with close frequencies in some modes with a mixture of their properties, namely discontinuities or jumps around certain magnetic surfaces (such as in pure Alfven waves), global spatial distribution of the normal velocity component and non-zero density perturbations (such as in fast waves).

High resolution observations of the solar corona made with instruments onboard the SOHO and TRACE spacecrafts have provided new evidence for the presence of oscillations in a variety of coronal magnetic structures. Most of these observations have been interpreted in terms of linear standing or propagating magnetohydrodynamic (MHD) waves, but the theoretical understanding is far from complete since analytical solutions to the linearised MHD wave equations can only be found for very simple magnetic congurations. Taking into account that the solar corona is basically structured by force-free magnetic elds, our purpose in this paper is to present the derivation of the linear MHD wave equations for a two-dimensional force-free magnetic eld conguration having longitudinal invariance, as well as to introduce a numerical code to solve the resulting system of coupled partial dierential equations. The accuracy of the code has been checked by numerically solving two cases for which analytical or simple numerical solutions exist. To our knowledge, this is the only two-dimensional code developed to study the normal MHD modes of oscillation of a general force-free eld with longitudinal invariance.

High-resolution observations provide evidence of the existence of small-amplitude transverse oscillations in solar filament fine structures. These oscillations are believed to represent fast magnetohydrodynamic (MHD) waves and the disturbances are seen to be damped in short timescales of the order of 1–4 periods. We propose that, due to the highly inhomogeneous nature of the filament plasma at the fine-structure spatial scale, the phenomenon of resonant absorption is likely to operate in the temporal attenuation of fast MHD oscillations. By considering transverse inhomogeneity in a straight flux tube model we find that, for density inhomogeneities typical of filament threads, the decay times are of a few oscillatory periods only.

During last years, direct evidence about oscillations in different coronal structures has been obtained thanks to the detailed observations made by SoHO and TRACE. With the help of magnetohydrodynamic (MHD) wave theory, we can explain these observations as due to the excitation and propagation of MHD waves in the solar corona. In spite that there are many solar coronal structures in which oscillations have been detected (prominences, loops, plumes, coronal holes, etc.), in the following we will concentrate on transverse oscillations of coronal loops, reviewing some theoretical models developed to understand these oscillations in terms of MHD waves.